#181308

order	512
nilpotency class	3
isoclinism	isoclinic class of #7943
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#95
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2,4,8,5,6,3,7)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23);(2,6)(3,4)(9,11,13,20,14,22,12,21)(10,15,16,23,18,24,17,19);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,19,24,23);(9,12,14,13)(10,16,18,17)(11,21,22,20)(15,23,24,19)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,e^2=hi,f^2=i,g^2=hi,b^a=be,e^a=eh,g^a=ghi,c^b=ch,f^b=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88504);